Calculus
7th Edition
ISBN: 9781524916817
Author: SMITH KARL J, STRAUSS MONTY J, TODA MAGDALENA DANIELE
Publisher: Kendall Hunt Publishing
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Chapter 13.3, Problem 27PS
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9. ∇⋅F of F = (2x3y)i-(3y2z)j+(xz3)k at (1,2,-1).
a.
27
b.
-34
c.
4
d.
-7
4. The vector field (3r²y + y +2, x³ + x − 1) is conservative. Find a function f such that Vƒ = F.
7. Sketch the vector field F = (x − y)î + yĵ. Identify where F₁ vanishes. Identify where
F2 vanishes. Plot a few flow lines.
8. Repeat the previous problem for F = yî + (x − y)î.
Chapter 13 Solutions
Calculus
Ch. 13.1 - Prob. 1PSCh. 13.1 - Prob. 2PSCh. 13.1 - Prob. 3PSCh. 13.1 - Prob. 4PSCh. 13.1 - Prob. 5PSCh. 13.1 - Prob. 6PSCh. 13.1 - Prob. 7PSCh. 13.1 - Prob. 8PSCh. 13.1 - Prob. 9PSCh. 13.1 - Prob. 10PS
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- Let x=x(t) be a twice-differentiable function and consider the second order differential equation x+ax+bx=0(11) Show that the change of variables y = x' and z = x allows Equation (11) to be written as a system of two linear differential equations in y and z. Show that the characteristic equation of the system in part (a) is 2+a+b=0.arrow_forwardPlease help with the solution to the following two questions. This is Vector Analysis (Vector Calculus). They have to deal with conservative fields.arrow_forward8. For the vector field F= 2(x+y) -ŷ 2x² + 2xy, evaluate fF.ds where S is the upper hemisphere evaluate f (VxF) ds where of x² + y² +2²=1, and S is the upper hemisphere of x² + y² + 2² = 1. S Zarrow_forward
- Suppose we have a vector field Ē(x, y, z) defined by E = -Vộ. Prove or disprove the following assertion: V(V · Ē) = v²Ẽ (4) It might help to write out the components of E in terms of derivatives of ø.arrow_forwardProblem B. For the vector field F(x, y) = 2y + 4x 7, i evaluate fF.dr over the line segment C from P(2,−4) to Q(1,2). JFarrow_forward6. Compute fF dr where f(x, y) = (y² cos x, 2y sinx) and C is the straight line path starting at (1,1) and ending at (1,3). Hint: is the vector field conservative? (Answer: 8 sin 1)arrow_forward
- Is F1 = (yz + ex-y) i + (xz − e−)j+xyk a conservative vector field? If so, give its potential function; if not, explain why not. Is F2 = cos zi+sin zj - (y cos z + sin z) k a conservative vector field? If so, give its potential function; if not, explain why not. Of the two previous parts of this problem, exactly one should have been a conservative vector field. (If not, redo them.) Compute the integral Fi.dr where F; (either F1 or F2) is that conservative vector field, and C is the curve parameterized by r(t)=(t+sin 10t, -t + sin 10t, 10 - t2) where t = [−, π].arrow_forward5. Let F(r, y) = ye²"i+ (re" – 1) j. (a) Show that F is a conservative vector field. (b) Find a potential function for Farrow_forward3. Let f : R" →R be differentiable and fix ro E R". Assume Vf(ro) + 0, where Vf(ro) is the gradient vector of f at ro. Show that for any v e R" with |v| = 1, we have -|Vf(ro)| < D, f(xo) <|Vf(xo)]; with equality if and only if Vf(ro) |V{(ro)| v = +arrow_forward
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